27 research outputs found
Algorithmic and enumerative aspects of the Moser-Tardos distribution
Moser & Tardos have developed a powerful algorithmic approach (henceforth
"MT") to the Lovasz Local Lemma (LLL); the basic operation done in MT and its
variants is a search for "bad" events in a current configuration. In the
initial stage of MT, the variables are set independently. We examine the
distributions on these variables which arise during intermediate stages of MT.
We show that these configurations have a more or less "random" form, building
further on the "MT-distribution" concept of Haeupler et al. in understanding
the (intermediate and) output distribution of MT. This has a variety of
algorithmic applications; the most important is that bad events can be found
relatively quickly, improving upon MT across the complexity spectrum: it makes
some polynomial-time algorithms sub-linear (e.g., for Latin transversals, which
are of basic combinatorial interest), gives lower-degree polynomial run-times
in some settings, transforms certain super-polynomial-time algorithms into
polynomial-time ones, and leads to Las Vegas algorithms for some coloring
problems for which only Monte Carlo algorithms were known.
We show that in certain conditions when the LLL condition is violated, a
variant of the MT algorithm can still produce a distribution which avoids most
of the bad events. We show in some cases this MT variant can run faster than
the original MT algorithm itself, and develop the first-known criterion for the
case of the asymmetric LLL. This can be used to find partial Latin transversals
-- improving upon earlier bounds of Stein (1975) -- among other applications.
We furthermore give applications in enumeration, showing that most applications
(where we aim for all or most of the bad events to be avoided) have many more
solutions than known before by proving that the MT-distribution has "large"
min-entropy and hence that its support-size is large
Commutative Algorithms Approximate the LLL-distribution
Following the groundbreaking Moser-Tardos algorithm for the Lovasz Local
Lemma (LLL), a series of works have exploited a key ingredient of the original
analysis, the witness tree lemma, in order to: derive deterministic, parallel
and distributed algorithms for the LLL, to estimate the entropy of the output
distribution, to partially avoid bad events, to deal with super-polynomially
many bad events, and even to devise new algorithmic frameworks. Meanwhile, a
parallel line of work, has established tools for analyzing stochastic local
search algorithms motivated by the LLL that do not fall within the Moser-Tardos
framework. Unfortunately, the aforementioned results do not transfer to these
more general settings. Mainly, this is because the witness tree lemma,
provably, no longer holds. Here we prove that for commutative algorithms, a
class recently introduced by Kolmogorov and which captures the vast majority of
LLL applications, the witness tree lemma does hold. Armed with this fact, we
extend the main result of Haeupler, Saha, and Srinivasan to commutative
algorithms, establishing that the output of such algorithms well-approximates
the LLL-distribution, i.e., the distribution obtained by conditioning on all
bad events being avoided, and give several new applications. For example, we
show that the recent algorithm of Molloy for list coloring number of sparse,
triangle-free graphs can output exponential many list colorings of the input
graph
LIPIcs
The LovĂĄsz Local Lemma (LLL) is a powerful tool in probabilistic combinatorics which can be used to establish the existence of objects that satisfy certain properties. The breakthrough paper of Moser and Tardos and follow-up works revealed that the LLL has intimate connections with a class of stochastic local search algorithms for finding such desirable objects. In particular, it can be seen as a sufficient condition for this type of algorithms to converge fast. Besides conditions for existence of and fast convergence to desirable objects, one may naturally ask further questions regarding properties of these algorithms. For instance, "are they parallelizable?", "how many solutions can they output?", "what is the expected "weight" of a solution?", etc. These questions and more have been answered for a class of LLL-inspired algorithms called commutative. In this paper we introduce a new, very natural and more general notion of commutativity (essentially matrix commutativity) which allows us to show a number of new refined properties of LLL-inspired local search algorithms with significantly simpler proofs
Using deep learning to construct stochastic local search SAT solvers with performance bounds
The Boolean Satisfiability problem (SAT) is the most prototypical NP-complete
problem and of great practical relevance. One important class of solvers for
this problem are stochastic local search (SLS) algorithms that iteratively and
randomly update a candidate assignment. Recent breakthrough results in
theoretical computer science have established sufficient conditions under which
SLS solvers are guaranteed to efficiently solve a SAT instance, provided they
have access to suitable "oracles" that provide samples from an
instance-specific distribution, exploiting an instance's local structure.
Motivated by these results and the well established ability of neural networks
to learn common structure in large datasets, in this work, we train oracles
using Graph Neural Networks and evaluate them on two SLS solvers on random SAT
instances of varying difficulty. We find that access to GNN-based oracles
significantly boosts the performance of both solvers, allowing them, on
average, to solve 17% more difficult instances (as measured by the ratio
between clauses and variables), and to do so in 35% fewer steps, with
improvements in the median number of steps of up to a factor of 8. As such,
this work bridges formal results from theoretical computer science and
practically motivated research on deep learning for constraint satisfaction
problems and establishes the promise of purpose-trained SAT solvers with
performance guarantees.Comment: 15 pages, 9 figures, code available at
https://github.com/porscheofficial/sls_sat_solving_with_deep_learnin
A new notion of commutativity for the algorithmic Lov\'{a}sz Local Lemma
The Lov\'{a}sz Local Lemma (LLL) is a powerful tool in probabilistic
combinatorics which can be used to establish the existence of objects that
satisfy certain properties. The breakthrough paper of Moser and Tardos and
follow-up works revealed that the LLL has intimate connections with a class of
stochastic local search algorithms for finding such desirable objects. In
particular, it can be seen as a sufficient condition for this type of
algorithms to converge fast.
Besides conditions for existence of and fast convergence to desirable
objects, one may naturally ask further questions regarding properties of these
algorithms. For instance, "are they parallelizable?", "how many solutions can
they output?", "what is the expected "weight" of a solution?", etc. These
questions and more have been answered for a class of LLL-inspired algorithms
called commutative. In this paper we introduce a new, very natural and more
general notion of commutativity (essentially matrix commutativity) which allows
us to show a number of new refined properties of LLL-inspired local search
algorithms with significantly simpler proofs
Improved Bounds for Randomly Colouring Simple Hypergraphs
We study the problem of sampling almost uniform proper q-colourings in k-uniform simple hypergraphs with maximum degree ?. For any ? > 0, if k ? 20(1+?)/? and q ? 100?^({2+?}/{k-4/?-4}), the running time of our algorithm is O?(poly(? k)? n^1.01), where n is the number of vertices. Our result requires fewer colours than previous results for general hypergraphs (Jain, Pham, and Vuong, 2021; He, Sun, and Wu, 2021), and does not require ?(log n) colours unlike the work of Frieze and Anastos (2017)
The Moser-Tardos Framework with Partial Resampling
The resampling algorithm of Moser \& Tardos is a powerful approach to develop
constructive versions of the Lov\'{a}sz Local Lemma (LLL). We generalize this
to partial resampling: when a bad event holds, we resample an
appropriately-random subset of the variables that define this event, rather
than the entire set as in Moser & Tardos. This is particularly useful when the
bad events are determined by sums of random variables. This leads to several
improved algorithmic applications in scheduling, graph transversals, packet
routing etc. For instance, we settle a conjecture of Szab\'{o} & Tardos (2006)
on graph transversals asymptotically, and obtain improved approximation ratios
for a packet routing problem of Leighton, Maggs, & Rao (1994)
Algorithms and Generalizations for the Lovasz Local Lemma
The Lovasz Local Lemma (LLL) is a cornerstone principle of the probabilistic method for combinatorics. This shows that one can avoid a large of set of âbad-eventsâ (forbidden configurations of variables), provided the local conditions are satisfied. The original probabilistic formulation of this principle did not give efficient algorithms. A breakthrough result of Moser & Tardos led to an framework based on resampling
variables which turns nearly all applications of the LLL into efficient algorithms. We extend and generalize the algorithm of Moser & Tardos in a variety of ways.
We show tighter bounds on the complexity of the Moser-Tardos algorithm, particularly its parallel form. We also give a new, faster parallel algorithm for the LLL.
We show that in some cases, the Moser-Tardos algorithm can converge even thoughthe LLL itself does not apply; we give a new criterion (comparable to the LLL) for determining when this occurs. This leads to improved bounds for k-SAT and hypergraph coloring among other applications.
We describe an extension of the Moser-Tardos algorithm based on partial resampling, and use this to obtain better bounds for problems involving sums of independent random variables, such as column-sparse packing and packet-routing. We describe a variant of the partial resampling algorithm specialized to approximating column-sparse covering integer programs, a generalization of set-cover. We also give hardness reductions and integrality gaps, showing that our partial resampling based algorithm obtains nearly optimal approximation factors.
We give a variant of the Moser-Tardos algorithm for random permutations, one of the few cases of the LLL not covered by the original algorithm of Moser & Tardos. We use this to develop the first constructive algorithms for Latin transversals and hypergraph packing, including parallel algorithms.
We analyze the distribution of variables induced by the Moser-Tardos algorithm. We show it has a random-like structure, which can be used to accelerate the Moser-Tardos algorithm itself as well as to cover problems such as MAX k-SAT in which we only partially avoid bad-events
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Stochastic Local Search and the Lovasz Local Lemma
Stochastic Local Search and the Lovasz Local LemmabyFotios IliopoulosDoctor of Philosophy in Computer ScienceUniversity of California, BerkeleyProfessor Alistair Sinclair, ChairThis thesis studies randomized local search algorithms for finding solutions of constraint satisfaction problems inspired by and extending the Lovasz Local Lemma (LLL). The LLL is a powerful probabilistic tool for establishing the existence of objects satisfying certain properties (constraints). As a probability statement it asserts that, given a family of âbadâ events, if each bad event is individually not very likely and independent of all but a small number of other bad events, then the probability of avoiding all bad events is strictly positive. In a celebrated breakthrough, Moser and Tardos made the LLL constructive for any product probability measure over explicitly presented variables. Specifically, they proved that whenever the LLL condition holds, their Resample algorithm, which repeatedly selects any occurring bad event and resamples all its variables according to the measure, quickly converges to an object with desired properties. In this dissertation we present a framework that extends the work of Moser and Tardos and can be used to analyze arbitrary, possibly complex, focused local search algorithms, i.e., search algorithms whose process for addressing violated constraints, while local, is more sophisticated than obliviously resampling their variables independently of the current configuration. We give several applications of this framework, notably a new vertex coloring algorithm for graphs with sparse vertex neighborhoods that uses a number of colors that matches the algorithmic barrier for random graphs, and polynomial time algorithms for the celebrated (non-constructive) results of Kahn for the Goldberg-Seymour and List-Edge-Coloring Conjectures.Finally, we introduce a generalization of Kolmogorovâs notion of commutative algorithms, cast as matrix commutativity, and show that their output distribution approximates the so-called âLLL-distributionâ, i.e., the distribution obtained by conditioning on avoiding all bad events. This fact allows us to consider questions such as the number of possible distinct final states and the probability that certain portions of the state space are visited by a local search algorithm, extending existing results for the Moser-Tardos algorithm to commutative algorithms